We prove that if is a sequence of bounded degree trees such that has vertices, then has a decomposition into . This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.
Cite this article
Felix Joos, Jaehoon Kim, Daniela Kühn, Deryk Osthus, Optimal packings of bounded degree trees. J. Eur. Math. Soc. 21 (2019), no. 12, pp. 3573–3647DOI 10.4171/JEMS/909